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Problem 6.41
Given:
Consider two adjacent plane quadratic elements, as shown.
Show that the shape functions
provide interelement continuity of the field quanity along the boundary.
Solution:
Along the right edge of the left element:
ξ
1
:=
N
2
1
4
1
ξ
+
()
⋅
1
η
−
⋅
1
4
1
ξ
2
−
⋅
1
η
−
⋅
−
1
4
1
ξ
+
⋅
1
η
2
−
⋅
−
=
N
2
η
2
2
η
2
−
=
→
N
3
1
4
1
ξ
+
⋅
1
η
+
⋅
1
4
1
ξ
+
⋅
1
η
2
−
⋅
−
1
4
1
ξ
2
−
⋅
1
η
+
⋅
−
=
N
3
η
2
2
η
2
+
=
→
N
6
1
2
1
ξ
+
⋅
1
η
2
−
⋅
=
N
6
1
η
2
−
=
→
let
ϕ
A
ϕ
2
=
ϕ
B
ϕ
3
=
ϕ
C
ϕ
6
=
ϕ
N
2
ϕ
A
⋅
N
3
ϕ
B
⋅
+
N
6
ϕ
C
⋅
+
=
η
2
η
−
2
ϕ
A
⋅
ηη
2
+
2
ϕ
B
⋅
+
1
η
2
−
ϕ
C
⋅
+
=
Along the left edge of the right element:
ξ
1
−
:=
N
1
1
4
1
ξ
−
⋅
1
η
−
⋅
1
4
1
ξ
2
−
⋅
1
η
−
⋅
−
1
4
1
ξ
−
⋅
1
η
2
−
⋅
−
=
N
1
η
2
2
η
2
−
=
→
N
4
1
4
1
ξ
−
⋅
1
η
+
⋅
1
4
1
ξ
2
−
⋅
1
η
+
⋅
−
1
4
1
ξ
−
⋅
1
η
2
−
⋅
−
=
N
4
η
2
2
η
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This note was uploaded on 01/11/2011 for the course MAE 5020 taught by Professor Folkman during the Fall '10 term at Utah State University.
 Fall '10
 Folkman

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